arXiv:hep-ph/0007250v1 24 Jul 2000
Vacuum properties in the presence of quantum fluctuations of the quark condensate.∗ Georges Ripka Service de Physique Th´eorique Centre d’Etudes de Saclay 91191 Gif-sur-Yvette, France
[email protected] July 14, 2000
Abstract The quantum fluctuations of the quark condensate are calculated using a regulated Nambu Jona-Lasinio model. The corresponding quantum fluctuations of the chiral fields are compared to those which are predicted by an ”equivalent” sigma model. They are found to be large and comparable in size but they do not restore chiral symmetry. The restoration of chiral symmetry is prevented by an ”exchange term” of the pion field which does not appear in the equivalent sigma model. A vacuum instability is found to be dangerously close when the model is regulated with a sharp 4-momentum cut-off.
1
Introduction.
This lecture discusses the modifications of vacuum properties which could arise due to quantum fluctuations of the chiral field, more specifically, due to the quantum fluctuations of the quark condensate. The latter is found to be Talk delivered at the workshop on Few-Quark Problems in Bled, Slovenia, July 8-16, 2000. ∗
1
surprisingly large, the root mean square deviation of the quark condensate attaining and exceeding 50% of the condensate itself. We shall discuss two distinct modifications of the vacuum: restoration of chiral symmetry due to quantum fluctuations of the chiral field, as heralded by Kleinert and Van den Boosche [1], and a vacuum instability not related to chiral symmetry restoration [2].
2 2.1
Chiral symmetry restoration due to quantum fluctuations of the chiral field. The linear sigma model argument.
The physical vacuum with a spontaneously broken chiral symmetry is often described by the linear sigma model, which, in the chiral limit (mπ = 0), has a euclidean action of the form: ! Z 2 1 κ2 2 1 2 2 σ + πi2 − fπ2 (1) (∂µ σ) + (∂µ πi ) + I = d4 x 2 2 8 Classically, we have (for translationally invariant fields): σ 2 + πi2 = fπ2
(2)
and the vacuum stationary point is: σ = fπ
πi = 0
(3)
We assume that κ2 is large enough (and the σ-meson is heavy enough) not to have to worry about the quantum fluctuations of the σ field. So we quantize the pion field while the σ field remains classical. We may then say that: σ 2 = fπ2 − hπi2 i. Classically, hπi2 i = 0 but the quantum fluctuations of the pion field make hπi2 i > 0 and therefore σ 2 < fπ2 . Let us estimate the fluctuation hπi2 i of the pion field. A system of free pions of mass mπ is described by the partition function: Z=
Z
− 12
D (π) e
R
2 +m2 π d4 xπi (−∂µ π) i
= e− 2 tr ln(−∂µ +mπ ) 1
2
2
(4)
It follows that: Z D E X ∂ ln Z ∂ 1 1 2 1 1 2 2 d4 x πi2 (x) = − = tr ln −∂ + m = Ω N − 1 µ π f 2 2 2 2 ∂mπ ∂mπ 2 2 k + m2π k1 (9) f fπ2 16π 2 fπ2 With fπ = 93 MeV and with Nf2 − 1 = 3 pions, the condition reads: Λ2 >
1 2.20 × 10−6
Λ > 674 MeV
(10)
In most calculations which use the Nambu Jona-Lasinio model, this condition is fulfilled. We conclude that the quantum fluctuations of the pion do indeed restore chiral symmetry. If we had used a 3-momentum cut-off, chiral symmetry would be restored when Λ > 477 MeV .
2.2
The non-linear sigma model argument.
We now argue that this is precisely what is claimed by Kleinert and Van den Boosche [1], although it is said in a considerably different language. They argue as follows. If κ2 (and therefore the σ mass) is large enough, the action can be thought of as the action of the non-linear sigma model, which in turn can be viewed as an action with Nf2 fields, namely (σ, πi ), subject to the constraint: σ 2 + πi2 = fπ2 (11)
3
The way to treat the non-linear sigma model is in the textbooks [3]. We work with the action: Iλ (σ, π) =
Z
1 1 d4 x (∂µ σ)2 + (∂µ πi )2 + λ σ 2 + πi2 − fπ2 2 2
(12)
in which we add a constraining parameter λ. The action is made stationary with respect to variations of λ. We integrate out the π field, to get the effective action: Iλ (σ) =
Z
d4 x
1 1 (∂µ σ)2 + λ σ 2 − fπ2 + tr ln −∂µ2 + λ 2 2
(13)
The action is stationary with respect to variations of λ and σ if: λσ = 0
σ 2 = fπ2 −
X 1 1 2 Nf − 1 2 2 k k +λ
(14)
So either λ = 0 and σ 6= 0, in which case we have: σ 2 = fπ2 −
X 1 1 2 Nf − 1 2 2 k k
(15)
or λ 6= 0 and σ = 0. The condition (15) is exactly the same as the condition (7). Thus, the hπ2 i ”stifness factor”, discussed in Ref.[1], is nothing but a measure of f i2 . π
3
Quantum fluctuations of the quark condensate calculated in the Nambu Jona-Lasinio model.
We now show that the quantum fluctuations of the chiral field are indeed large in the Nambu Jona-Lasinio model, but that chiral symmetry is far from being restored. The regularized Nambu Jona-Lasinio model is defined in section 4. We begin by giving some results. In the Nambu Jona-Lasinio model, the chiral field is composed of a scalar field S and Nf2 − 1 pseudoscalar fields Pi . They are related to the quark bilinears: ¯ ¯ 5 τi ψ S = V ψψ Pi = V ψiγ (16) 4
2
where V = − Ng c is the 4-quark interaction strength. The quark propagator in the vacuum is: 1 (17) kµ γµ + M0 rk2 and the model is regularized using either a sharp 4-momentum cut-off or a soft gaussian cut-off function: rk = 1 if k 2 < Λ2 rk = 0 if k > Λ (sharp cut − of f ) 2 − k2 2Λ
rk = e
(gaussian regulator)
(18) (19)
. Let ϕ0 = M0 be the strength of the scalar field in the physical vacuum. We shall show results obtained with typical parameters. If we choose M0 = 300 MeV and Λ = 750 MeV , then MΛ0 = 0.4. We then obtain fπ = 94.6 MeV with a sharp cut-off and fπ = 92.4 MeV with a gaussian cut-off (in the chiral limit). The interaction strengths are: V = −9.53 Λ−2 (sharp cut − of f )
V = −18.4 Λ−2 (gaussian cut − of f ) (20) and the squared pseudo-scalar field has the expection value D
Pi2
E
=V
2
2
(21)
Zπ M0
(22)
2
(23)
¯ 5 τi ψ ψiγ
At low q we identify the pion field as: πi =
q
Z π Pi
fπ =
so that, in the Nambu Jona-Lasinio model: hπi2 i fπ2 where
3.1
¯ 5 τi ψ ψiγ
2
V2 =
q
¯ 5 τi ψ ψiγ M02
is the pion contribution to the squared condensate.
Results obtained for the quark condensate and for the quantum fluctuations of the chiral field.
Let us examine the values of the quark condensates and of the quantum fluctuations of the chiral field calculated in the chiral limit. 5
• The quark condensate calculated with a sharp cut off is: ¯ ψψ
D
E1 3
= −0.352 × Λ = 263 MeV
(sharp cut − of f )
(24)
is about 25 % smaller when it is calculated with a soft gaussian regulator: D
¯ ψψ
E1 2
= −0.280 × Λ = 210 MeV
(gaussian regulator)
(25)
• The magnitude of the quantum fluctuations of the pion field can be measured by the mean square deviation ∆2 of the condenstate from its classical value: 2 D E 2 ¯ ¯ 2 ∆ = ψΓa ψ − ψψ (26) The relative root mean square fluctuation of the condensate ∆ is: ∆ D E ¯ ψψ
= 0.41
(sharp cut − of f )
D
∆ E = 0.77 ¯ ψψ
(gaussian regulator)
(27) These are surprisingly large numbers, certainly larger than 1/Nc . The linear sigma model estimate did give us a fair warning that this might occur. hπi2 i fπ2
V2
D
¯ 5 τi ψ) (ψiγ
2
E
• This feature also applies to the ratio which was so = M02 crucial for the linear sigma model estimate of the restoration of chiral symmetry. We find: hπi2 i = 0.38 (sharp cut − of f ) fπ2
hπi2 i = 0.85 fπ2
(gaussian regulator) (28)
• In spite of these large quantum fluctuations of the chiral field, the quark condensates change by barely a few percent. This is shown in tables 1 and 2 where various contributions to the quark condensate are given in units of Λ3 . The change in the quark condensate is much smaller than 1/Nc .
6
¯ ψψ σ-contribution classical -0.04187 exchange term 0.00158 ring graphs 0.00014 total contribution -0.04015 D
E
π-contribution 0 -0.00475 0.00134 -0.00341
total -0.04187 -0.00317 0.00148 -0.04356
Table 1: Various contributions to the quark condensate calculated with a sharp 4-momentum cut-off and with M0 /Λ = 0.4. The quark condensate is expressed in units of Λ3 . ¯ ψψ σ-contribution classical -0.02178 exchange term 0.00162 ring graphs 0.00077 total contribution -0.0193 D
E
π-contribution 0 -0.00486 0.00228 -0.00258
total -0.02178 -0.00324 0.00305 -0.02197
Table 2: Various contributions to the quark condensate calculated with a gaussian cut-off function and with M0 /Λ = 0.4. The quark condensate is expressed in units of Λ3 .
3.2
The effect and meaning of the exchange terms.
The tables 1 and 2 show that, among the 1/Nc corrections, the exchange terms dominate. The exchange and ring graphs are illustrated on figures 1 and 2. The way in which they arise is explained in section 4.1. The exchange graphs contribute 2-3 times more than the remaining ring graphs. Furthermore, the pion contributes about three times more to the condensate than the sigma, so that the sigma field contributes about as much to the exchange term as any one of the pions. However, the exchange term in the pion channel enhances the quark condensate instead of reducing it. As a result of this there is a very strong cancellation between the exchange terms and the ring graphs. This is why the sigma and pion loops contribute so little to the quark condensate. They increase the condensate by 4% when a sharp cut-off is used, and by 1% when a gaussian regulator is used. This is about ten times less than 1/Nc . The ring graphs reduce the condensate (in absolute value) in both the sigma and pion channels. This can be expected. Indeed, the ring graphs 7
promote quarks from the Dirac sea negative energy orbits (which contribute negative values to the condensate) to the positive energy orbits (which contribute positive values to the condensate). The net result is a positive contribution to the condensate which reduces the negative classical value. What then is the meaning of the exchange terms? The exchange terms have the special feature of belonging to first order perturbation theory (see figures 1 and 2). Their contribution to the energy is not due to a modification of the Dirac sea. It is simply the exchange term arising in the expectation value of the quark-quark interaction in the Dirac sea. However, the contribution of the exchange term to the quark condensate does involve q q¯ excitations. These excitations are due to a modification of the constituent quark mass which is expressed in terms of quark-antiquark excitations of the Dirac sea. The exchange term is modifying (increasing in fact) the constituent quark mass and therefore the value of fπ . These results suggest that, in order to reduce the Nambu Jona-Lasinio model to an equivalent sigma model, it might be better to include the exchange term in the constituent quark mass, which is another way of saying that, in spite of the 1/Nc counting rule, it may be better to do Hartree-Fock theory than Hartree theory. The exchange (Fock) term should be included in the gap equation. The direct (Hartree) term is, of course, included in the classical bosonized action. In the equivalent sigma model, fπ is proportional to the constituent quark mass. Failure to notice that that the constituent quark mass is altered by the exchange term is what lead Kleinert and Van den Boosche to conclude erronously in Ref.[1] that chiral symmetry would be restored in the Nambu Jona-Lasinio model. They were right however in expecting large quantum fluctuations of the quark condensate.
4
The regularized Nambu Jona-Lasinio model.
The condensates quoted in section 3.1 were calculated with a regularized Nambu Jona-Lasinio model which is defined by the euclidean action:
Im (q, q¯) =
Z
2 ¯ − g + j ψΓ ¯ aψ 2 d4 x q¯ (−i∂µ γµ ) q + mψψ 2Nc
"
!
8
#
(29)
The euclidean Dirac matrices are γµ = γ µ = (iβ, ~γ ). The matrices Γa = (1, iγ5~τ ) are defined in terms of the Nf2 − 1 generators ~τ of flavor rotations. 2 Results are given for Nf = 2 flavors. The coupling constant Ng c is taken to be inversely proportional to Nc in order to reproduce the Nc counting rules. The current quark mass m is introduced D E as a source term used to calculate ¯ the regularized quark condensate ψψ . We have also introduced a source ¯ a ψ 2 which is used to calculate the squared quark condensate term 1 j ψΓ 2
¯ aψ ψΓ
2
.
The quark field is q (x) and the ψ (x) fields are delocalized quark fields, which are defined in terms of a regulator r as follows: ψ (x) =
Z
d4 x hx |r| yi q (y)
(30)
The regulator r is diagonal in k-space: hk |r| k ′ i = δkk′ r (k) and its explicit form in given in Eq.(18). The use of a sharp cut-off function is tantamount to the calculation of Feynman graphs in which the quark propagators are cut off at a 4-momentum Λ - a most usual practise. The regulator r, introduced by the delocalized fields, makes all the Feynman graphs converge. A regularization of this type results when quarks propagate in a vacuum described by in the instanton liquid model of the QCD (see Ref.[4] and further references therein). A Nambu Jona-Lasinio model regulated in this manner with a gaussian regulator was first used in Ref.[5], and further elaborated and applied in both the meson and the soliton sectors [5],[6],[7],[8], [9],[2]. Its properties are also discussed in [10]. With one exception. In this work, as in Ref.[2], the regulator multiplies the current quark mass. The introduction of the regulator in the current quark ¯ = m¯ mass term mψψ q r 2 q requires some explanation. The current quark D E ¯ mass m is used as a source term to calculate the quark condensate ψψ which, admittedly, would be finite (by reason of symmetry) even in the absence of a regulator - and, indeed, values of quark condensates are usually calculated with an unregularized source term in the Nambu Jona-Lasinio ac 2 D E2 ¯ aψ ¯ tion. However, when we calculate the fluctuation ψΓ − ψψ of the quark condensate, the expectation value
¯ aψ ψΓ
2
be inconsistent and difficult to interpret the fluctuation 9
diverges. It would
¯ aψ ψΓ
2
¯ − ψψ D
E2
¯ ¯ a ψ 2 using a if ψψ were evaluated using a bare source term and ψΓ ¯ = m¯ regulator. When a regularized source term mψψ q r 2 q is used, the current quark mass m can no longer be identified with the current quark mass term appearing in the QCD lagrangian. Of course, when a sharp cut-off is used, it makes no difference if the current quark mass term is multiplied by the regulator or not. We have seen in section 3.1 that the leading order D E ¯ 1/3 diminishes by only 20% when contribution to the quark condensate ψψ D
E
the sharp cut-off is replaced by a gaussian regulator. (This statement may be misleading because when the sharp cut-off is replaced by a gaussian regulator, the interaction strength V is also modified so as to fit fπ . If we use a gaussian regulator, D E the quark condensate calculated with a regulated source 2 ¯ term mr is ψψ = −0.0218 Λ3 whereas the quark condensate calculated D E ¯ = −0.0505 Λ3 .) with a bare source term m is ψψ The way in which the current quark mass of the QCD lagrangian appears in the low energy effective theory is model dependent and it has been studied in some detail in Ref.[11] within the instanton liquid model of the QCD vacuum [12],[13],[4]. An equivalent bosonized form of the Nambu Jona-Lasinio action (29) is:
1 (ϕ − m) (V − j)−1 (ϕ − m) (31) 2 The first term is the quark loop expressed in terms of the chiral field ϕ, which is a chiral 4-vector ϕa = (S, Pi ) so that ϕa Γa = S + iγ5 τi Pi . In the second term, the chiral 4-vector ma ≡ (m, 0, 0, 0) is the current quark mass and V is the local interaction: g2 hxa |V | ybi = − δab δ (x − y) (32) Nc Ij,m (ϕ) = −T r ln (−i∂µ γµ + rϕa Γa r) −
The partition function of the system, in the presence of the sources j and m is given by the expression: −W (j,m)
e
=
Z
1
D (ϕ) e−Ij,m (ϕ)− 2 tr ln(V −j)
¯ and the squared quark condensates The quark condensate ψψ D
E
(33)
¯ aψ ψΓ
2
can be calculated from the partition function W (j, m) using the expressions: E ¯ = 1 ∂W (j, m) ψψ Ω ∂m
D
2 1 ∂W (j, m) 1 ¯ ψΓa ψ =− 2 Ω ∂j
10
(34)
where Ω is the space-time volume d4 x = Ω. The stationary point ϕa = (M, 0, 0, 0) of the action is given by the gap equation: M (V − j)−1 = −4Nc Nf gM (35) M −m This equation relates the constituent quark mass M to the interaction strength V − j. R
4.1
The exchange and ring contributions.
The second order expansion of the action Ijm (ϕ) around the stationary point reads: 1 (36) Ijm (ϕ) = Ijm (M) + ϕ Π + (V − j)−1 ϕ 2 where Ijm (M) is the action calculated at the stationary point ϕ = (M, 0, 0, 0) and where Π is the polarization function (often referred to as the Lindhardt function): hxa |Π| ybi = −
δ T r ln (−i∂µ γµ + rϕa Γa r) δϕa (x) δϕb (y)
(37)
Substituting this expansion into the partition function (33), we can calculate the partition function W (j, m) using gaussian integration with the result: 1 W (j, m) = Ijm (M) + tr ln (1 − Π (V − j)) 2
(38)
The first term of the action (38) is what we refer to as the ”classical” action. The values labelled ”classical” in the tables displayed in section 3.1 are obtained by calculating the condensates (34) while retaining only the term Ijm in the partition function (38). The logarithm in (38) is what we refer to as the loop contribution. The expansion of the logarithm expresses the loop contribution in terms of the Feynman graphs shown on figure 1. The first term of the loop expansion is what we call the ”exchange term”, also referred to as the Fock term:1 : 1 Wexch = − trΠ (V − j) 2 The remaining terms are what we call the ring graphs. 11
(39)
+
+
+ ...
Figure 1: The Feynman graphs which represent the meson loop contribution to the partition function. The first graph is the exchange graph and the remaining graphs are the ring graphs. stop here It is simple to show that the inverse meson propagators are given by: K −1 = Π + (V − j)−1
(40)
They are diagonal in momentum and flavor space: hqa |K −1 | k ′ qi = δab δkk′ Ka (q) and a straighforward calculation yields the following explicit expressions for the S (sigma) and P (pion) inverse propagators: 1 2 22 M 26 44 (q) = 4Nc Nf (q) + fM (q) − gM (q) + q fM (q) + M 2 fM g(41) M 2 M −m 1 2 22 M 26 44 KP−1 (q) = 4Nc Nf (q) − fM (q) − gM (q) + q fM (q) + M 2 fM g(42) M 2 M −m
KS−1
where the loop integrals are: np fM
p n qr rk− 1X k+ 2q 2 (q) = Ω k q 2 q 2 4 4 2 2 k + 2 + rk+ q M k − 2 + rk− q M 2
(43)
2
and: 2 q rk− 1X 2 2 q gM (q) = rk+ 2 2 q Ω k k− 4 2 + rk− q M 2
gM ≡ gM (q = 0)
(44)
2
These are the expressions which are obtained from the second order expansion of the action (31) retaining the regulators from the outset and throughout. 12
+
+
Figure 2: The contribution to the quark condensate of the Feynman graphs ¯ The first graph shown on figure 1. The black blob represents the operator ψψ. (which is the dominating contribution) is the contribution of the exchange term. It represents q q¯ excitations which describe a change in mass of the Dirac sea quarks. This exchange graph would not appear in a Hartree-Fock approximation, which would include the exchange graph in the gap equation. Innumerable papers have been published (including some of my own) in which the meson propagators are derived from the unregulated Nambu Jona-Lasinio action: Ij,m (ϕ) = −T r ln (−i∂µ γµ + ϕa Γa ) −
1 (ϕ − m) (V − j)−1 (ϕ − m) 2
(45)
The expressions obtained for the propagators are then: 1 2 m q + 4M 2 fM (q) + gM 2 M −m m 1 2 q fM (q) + gM KP−1 (q) = 4Nc Nf 2 M −m
KS−1 (q) = 4Nc Nf
(46) (47)
where the loop integrals are: fM (q) =
1 X 1 2 q 2 Ω k 4 before it becomes apparent that the energy is not bounded from below. The instability is an unpleasant feature of effective theories which use 18
relatively low cut-offs. However, the low value of the cut-off is dictated by the vacuum properties and we need to learn to work with it. Further details are found in Ref.[2]. We conclude from this analysis that it is much safer to use a soft regulator, such as a gaussian, than a sharp cut-off.
References [1] H. Kleinert and B. Van Den Boosche. Phys.Lett. B474, page 336, 2000. [2] G.Ripka. Quantum fluctuations of the quark condensate. ph/0003201, 2000.
hep-
[3] J.Zinn-Justin. Quantum Field Theory and Critical Phenomena. Clarendon Press, Oxford, 1989. [4] C. Weiss D.I. Diakonov, M.V. Polyakov. Hadronic matrix elements of gluon operators in the instanton vacuum. Nucl. Phys. B461, page 539, 1996. [5] R.D.Bowler and M.C.Birse. Nucl.Phys. A582, page 655, 1995. [6] R.S.Plant and M.C.Birse. Nucl.Phys. A628, page 607, 1998. [7] W.Broniowski B.Golli and G.Ripka. Phys.Lett. B437, page 24, 1998. [8] Wojciech Broniowski. Mesons in non-local chiral quark models. hepph/9911204, 1999. [9] B.Szczerebinska and W.Broniowski. Acta Polonica 31, page 835, 2000. [10] Georges Ripka. Quarks Bound by Chiral Fields. Oxford University Press, Oxford, 1997. [11] M.Musakhanov. Europ.Phys.Journal C9, page 235, 1999. [12] E.Shuryak. Nucl.Phys. B203, pages 93,116,140, 1982. [13] D.I.Diakonov and V.Y.Petrov. Nucl.Phys. B272, page 457, 1986. [14] D.Blaschke S.Schmidt and Y.Kalinovsky. Phys.Rev. C50, page 435, 1994. 19
[15] S.P. Klevansky Y.B. He, J. Hfner and P. Rehberg. Nucl. Phys. A630, page 719, 1998. [16] E.N.Nikolov W.Broniowski, G.Ripka and K.Goeke. Zeit.Phys. A354, page 421, 1996.
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